Butter and flour a straight-walled 10-inch tube pan. Preheat the oven to 350 degrees F.
In a standing mixer fitted with the paddle attachment, cream the butter until light and fluffy. Add the sugar and continue to mix well. Add the eggs 1 at a time and beat until incorporated.
In a bowl, whisk together the milk and vanilla. In another bowl, whisk together the flour, baking powder, and salt.
Reduce the mixer's speed to low. Alternately add the flour and milk mixture until a smooth batter is formed. Continue mixing for 5 minutes. (It's important to mix for the full 5 minutes.)
Transfer 1/3 of the batter to a bowl and whisk in the cocoa powder. Stir the coconut into the remaining batter.
Pour the coconut batter into the pan; then spoon the chocolate batter around. Run a knife through the batter, cutting the batters together to create a marbled effect. (Do not stir.) Bake until a toothpick inserted into the middle of the cake comes out clean, about 1 hour 15 minutes.
Let the cake cool in the pan. Invert the cake onto a stand or plate, slice, and serve.